5.7 Eulerizing Graphs

Euler circuit and Euler path do not always exist. There are many graphs (in real life) that have more than 2 odd vertices. Instead of finding a route that travels along the edges of a graph and passes through each and every edge of the graph at least once, we want to find a a route that re- cross the fewest number of edges.

Eulerizing of a graph Eulerizing is the process of changing all odd vertices to even vertices by duplicating appropriate edges

Eulerizing Graphs First step is to identify the odd vertices. Second step is to add duplicate copies of edges to create all even vertices OPTIMAL ROUTE: duplicate the fewest number of edges NOT an optimal route illegal route

Eulerizing the following graphs 1) 2)

Semi-eulerizing of a graph Semi-eulerizing is the process of leaving 2 odd vertices on the graph unchanged and changing other odd vertices to even vertices.

Semi-eulerizing Graphs First step is to identify the odd vertices. Second step is leave out 2 odd vertices and add duplicate copies of edges to create even vertices OPTIMAL ROUTE:dup licate the fewest number of edges NOT an optimal route illegal route

Semi eulerizing the following graphs 1) 2)